Inference, Explanation, and Other Frustrations: Essays in the Philosophy of Science [Reprint 2019 ed.] 9780520309876

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Inference, Explanation, and Other Frustrations

Pittsburgh Series ili Philosophy and History of Science

Series Editors: Adolf Griinbaum Larry Laudan Nicholas Rescher Wesley C. Salmon

Inference, Explanation, and Other Frustrations Essays in the Philosophy of Science

E D I T E D BY

J o h n Earman

U N I V E R S I T Y OF CALIFORNIA PRESS Berkeley Los Angeles Oxford

University of California Press Berkeley and Los Angeles, California University of California Press Oxford, England Copyright © 1992 by T h e Regents of the University of California Library of Congress Cataloging-in-Publication Data Inference, explanation, and other frustrations : essays in the philosophy of science / edited by J o h n E a r m a n . p. cm. — (Pittsburgh series in philosophy and history of science; v. 14) "Papers delivered in the a n n u a l lecture series (1986-1989) sponsored by the University of Pittsburgh's Center for the Philosophy of Science."—Intro. Includes bibliographical references and index. ISBN 0-520-07577-3 (alk. paper) ISBN 0-520-08044-0 (pbk.: alk. paper) 1. Science—Philosophy. 2. Science—Methodology. 3. Inference. 4. Induction (Logic) I. E a r m a n , J o h n . II. Series. Q.175.3.I53 1992 501—dc20 91-40044 CIP Printed in the United States of America 1 2 3 4 5 6 7 8 9 T h e paper used in this publication meets the minimum requirements of American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ANSI Z 3 9 . 4 8 - 1 9 8 4 ©

CONTENTS

INTRODUCTION

/

vii

PART I • INFERENCE AND M E T H O D

/

/

1. Thoroughly M o d e r n Meno Clark Glymour and Kevin Kelly / 3 1. The Concept of Induction in the Light of the Interrogative Approach to Inquiry Jaakko Hintikka / 23 3. Aristotelian Natures and the Modern Experimental Method Nancy Cartwright / 44 4. Genetic Inference: A Reconsideration of David Hume's Empiricism Barbara D. Massey and Gerald J. Massey / 72 5. Philosophy and the Exact Sciences: Logical Positivism as a Case Study Michael Friedman / 84 6. Language and Interpretation: Philosophical Reflections and Empirical Inquiry Noam Chomsky / 99

PART I I - T H E O R I E S AND EXPLANATION 7. Constructivism, Realism, and Philosophical Method Richard Boyd / 131 8. Do We Need a Hierarchical Model ofScience? Diderik Batens / 199 9. Theories of Theories: A View from Cognitive Science Richard E. Grandy / 216 v

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129

VI

CONTENTS

10. Procedural Syntax for Theory Elements Joseph D. Sneed / 234 11. Why Functionalism Didn't Work Hilary Putnam / 255 12. Physicalism Hartry Field / 271 CONTRIBUTORS

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INDEX

295

I

293

INTRODUCTION

T h e present volume contains papers delivered in the twenty-seventh, twentyeighth, and twenty-ninth annual Lecture Series (1986-1989) sponsored by the University of Pittsburgh's Center for the Philosophy o f Science. T h e authors will be immediately recognized as among the leading lights in current philosophy of science. Thus, taken together, the papers provide a good sample of work being done at the frontiers o f research in philosophy o f science. T h e y illustrate both the contemporary reassessment of our philosophical heritages and also the opening of new directions o f investigation. T h e brief remarks that follow cannot hope to d o justice to the rich and rewarding fare to be found herein but are supposed to serve only as a menu.

INFERENCE AND M E T H O D Students in philosophy of science used to be taught to respect the distinction between " t h e context of d i s c o v e r y " and " t h e context of justification." T h e philosophy of science (so the story w e n t ) is concerned with the latter context but not the former. It seeks to provide principles for evaluating scientific hypotheses and theories once they are formulated, but it must remain modestly silent about the process o f discovery since hypotheses and theories are free creations o f the human mind and since the creative process is the stuff o f psychology, not philosophy. T h e discovery/justification distinction is now under pressure from several directions, one o f which stems from work in artificial intelligence and formal learning theory. Granted that scientists d o in fact arrive at theories by a process o f guesswork, intuition, or whatever, it remains to ask what true theories can be reliably discovered by what procedures. M o r e specifically, for a specified kind of theory and a specified class o f possible vii

MU

INTRODUCTION

worlds, does there exist a procedure (recursive or otherwise) such that for every possible evidence sequence from any of the possible worlds the procedure eventually finds every true theory of the given type and eventually avoids every false theory of the given type? In their contribution, Clark Glymour and Kevin Kelly show how to make such questions precise, and for some precise versions they provide precise answers. But as they note, a host of such questions remain begging for further investigation. J a a k k o Hintikka's contribution draws out some of the implications for induction of his interrogative model of inquiry. This model conceptualizes scientific inquiry as a game played by a scientist against Nature. T h e scientist's goal is to derive a conclusion C from a starting premise P. T o reach this goal, the scientist is allowed two kinds of moves: an interrogative move in which a question is put to Nature and an answer received, and a deductive move in which he draws logical consequences from P and the answers received to interrogative moves. A very striking feature of this model is the absence of any place for induction as it is traditionally conceived. Hintikka argues that Hume's classic problem of induction is an artifact of the mistaken assumption that the only answers N a t u r e gives to queries are in the form of atomic (i.e., quantifierfree) sentences. Hintikka sides with the view, traceable to Newton and beyond Newton to Aristotle, that observation and experiment provide us with propositions that possess a significant generality. T h e residual, non-Humean problem of induction, as Hintikka conceives it, consists in extending the scopes of and unifying the general truths received from Nature. According to the textbooks, modern science eschews Aristotelian natures in favor of laws of nature construed as codifications of regularities. In her provocative contribution Nancy Cartwright contends that this common wisdom is flawed, for in her view laws of nature are about natures. Thus, for Cartwright, Newton's law of gravitation doesn't say what forces bodies actually experience but rather what forces it is their nature, as massive objects, to experience. T h e exceptionless regularities required by the empiricist account are rarely found, she contends, and where they are found they result from arrangements that allow stable natures to be manifested. Cartwright supports her neo-Aristotelian conception of laws by arguing that it makes more sense of experimental methodology and inductive procedures than the more popular empiricist view. If empiricism is the view that no matter of fact can be known a priori, then H u m e was not an empiricist. For, as Barbara and Gerald Massey show in their contribution, Hume's account of animals attributes to them factual knowledge which is not learned from experience but which is imparted to them by "the original hand of Nature." H u m e could be said to remain an empiricist insofar as he denies that h u m a n beings have specialized innate cognitive faculties or instincts as opposed to generalized instincts, such as the inductive propensity. But the distinction between specialized and generalized propensities is vague and, thus, the boundaries of empiricism are fuzzy. If Nelson Goodman is right,

INTRODUCTION

IX

we are endowed with the propensity to project 'green' instead of 'grue.' And Noam Chomsky has championed the view that we are endowed with complex propensities to m a p linguistic evidence to linguistic knowledge. D o such propensities, which are at once special and general, lie inside or outside the boundaries of empiricism? Logical positivism is a failed program. But its real shortcomings are quite different from those besetting the caricatures that dot the potted histories of philosophy. For example, the leading logical positivists (apart from Schlick) did not subscribe to the naive empiricism of a neutral observation language; indeed, as Michael Friedman notes in his contribution, the theory-ladenness of observation was explicitly emphasized by Carnap and others. Friedman argues that the ultimate shortcoming of positivism as embodied, say, in Carnap's Logical Syntax of Language lay in its failure to establish a neutral framework from which alternative languages or frameworks could be judged. Friedman traces this failure to Godel's incompleteness theorems and argues that the demise of Carnap's program does not promote relativism—as expressed by a notion of truth relativized to a framework—but pulls the rug out from under this and other fashionable relativisms. I f asked to list the most important accomplishments of twentieth-century philosophy, the majority of the profession would surely give prominent place to Quine's slaying of one of two dogmas of empiricism—the existence of the analytic-synthetic distinction (that is, a principled distinction between truths of meaning and truths of fact). This accomplishment would not appear on Noam Chomsky's list. Indeed, in his paper for this volume, Chomsky argues that Quine's result is, ironically, an artifact of an overly behavioristic and a too narrowly positivistic conception of how the scientific investigation of language should and does proceed. In particular, he claims that the strictures imposed by Quine's paradigm of "radical translation" are not accepted in and would undermine the process of inquiry in the natural sciences.

T H E O R I E S AND E X P L A N A T I O N T h e theories of modern science tell stories of unobservable entities and processes. Scientific realists contend that these stories are not to be read as fairy tales and that observational and experimental evidence favorable to a theory is to be taken as evidence that the theory gives us a literally true picture of the world. Richard Boyd, one of the leading exponents of scientific realism, has in the past been concerned to combat the logical empiricists and their heirs who (with some notable exceptions such as Hans Reichenbach) contend that scientific theories are to be read instrumentally or else that we are never warranted in accepting a theory except as being adequate to saving the phenomena. Here Boyd is concerned with the more elusive and insidious opponent of realism who contends that the very notion of " t h e world" to which theories can

X

INTRODUCTION

succeed or fail in corresponding is a delusion since science is the social construction of reality. S o m e forms of constructivism h a v e been successfully answered; for example, those t h a t take their cue f r o m K u h n i a n incommensurability c a n be rejected on the basis of a causal theory of reference. O t h e r m o r e subtle forms of constructivism r e m a i n to be answered. Boyd's c o n t r i b u t i o n is aimed at identifying t h e most interesting of these forms a n d showing t h a t the "philosophical p a c k a g e " in which they come w r a p p e d c a n n o t be reconciled with the c o n t e n t a n d p r o c e d u r e s of science. Diderik Batens gives a resounding " N o " to his q u e r y " D o W e Need a Hierarchical M o d e l of Science?" In place of both hierarchical a n d holistic models he proposes a contextualistic a p p r o a c h in which problems are always f o r m u lated a n d a t t a c k e d with respect to a localized problem-solving situation r a t h e r t h a n with respect to the full-knowledge situation. O n Batens's a c c o u n t , m e t h odological rules as well as empirical assertions are contextual. T h i s has the interesting consequence t h a t n o a priori a r g u m e n t s can d e m o n s t r a t e the superiority of science to astrology; r a t h e r the superiority has to be shown on a case-by-case basis in a r a n g e of concrete problem-solving contexts. W h a t was once the "received view" of scientific theories, which emphasized the representation of scientific theories as a logically closed set of sentences (usually in a first-order l a n g u a g e ) , has given way to a " s e m a n t i c " or " s t r u c t u r alist" view, e x p o u n d e d in different versions by Patrick Suppes, J o s e p h Sneed, Fredrick S u p p e , Bas v a n Fraassen, a n d others. But w h a t exactly is the difference between these two ways of u n d e r s t a n d i n g theories? A n d w h a t exactly was w r o n g with or lacking in the older view? In his c o n t r i b u t i o n R i c h a r d G r a n d y argues t h a t the p r o p o n e n t s of the semantic view are offering not so m u c h a new account of theories per se as a new account of the epistemology a n d application of theories. In his contribution Sneed responds to critics w h o c h a r g e that the semantic-structuralist reconstructions of theories are i n a d e q u a t e because they fail to provide syntactic representations of crucial items. By providing syntactic formulations of "lawlikeness," "theoretical concepts," a n d "constraints," Sneed paves the way for a reconciliation of the old a n d new views of theories, a n d at the same time he opens u p a new a v e n u e of research by connecting his structuralist account with previous work on d a t a bases. In his contribution Hilary P u t n a m explains why he has a b a n d o n e d a view he helped to articulate a n d p o p u l a r i z e — t h e c o m p u t a t i o n a l or functional c h a r acterization of the mental. H e continues to hold t h a t m e n t a l states c a n n o t be straightforwardly identified with physical states of the brain. But he now proposes to t u r n the tables on his former self by extending his own a r g u m e n t s , previously deployed to show t h a t " s o f t w a r e " is more i m p o r t a n t t h a n " h a r d w a r e , " to show that m e n t a l states are not straightforwardly identical with c o m p u t a t i o n a l states of the brain. W h a t does P u t n a m propose as a replacement for functionalism? S o m e hints are to be found in the present p a p e r a n d in his book Representation and Reality ( C a m b r i d g e , Mass.: M I T Press, 1988),

INTRODUCTION

XI

but for a complete answer the reader will have to stay tuned for further developments. Hartry Field is more sanguine about another major " i s m " — p h y s i c a l i s m . H e tries to chart a course between the Scylla of formulating the doctrine in such a strong form as to make it wholly implausible and the Charybdis of making it so weak as to have no methodological bite. T h e form of physicalism that Field takes to be worthy of respect is along the lines of reductionism, asserting (very roughly) that all good explanation must be reducible to physical explanation. H e argues that weaker versions of physicalism, such as supervenience, that lack the explanatory requirement founder on the Charybdis. What remains to be specified to make physicalism a definite thesis is the reduction base: what are the considerations in virtue of which a science or a theory is properly classified as being part of physics? John Earman University of Pittsburgh

ONE

Thoroughly Modern Meno Clark Glymour

1.

and Kevin

Kelly

INTRODUCTION

T h e Meno presents, and then rejects, an argument against the possibility of knowledge. T h e argument is given by M e n o in response to Socrates' proposal to search for what it is that is virtue: M e n o : How will you look for it, Socrates, when you do not know at all w h a t it is? H o w will you aim to search for something you do not know at all? I f you should meet with it, how will you know that this is the thing that you did not know? 1

M a n y commentators, including Aristotle in the Posterior Analytics, take Meno's point to concern the recognition of an object, and if that is the point there is a direct response: one can recognize an object without knowing all about it. But the passage can also be understood straightforwardly as a request for a discernible mark of truth, and as a cryptic argument that without such a mark it is impossible to acquire knowledge from the instances that experience provides. W e will try to show that the second reading is of particular interest. I f there is no mark of truth, nothing that can be generally discerned that true and only true propositions bear, Meno's remarks represent a cryptic argument that knowledge is impossible. W e will give an interpretation that makes the argument valid; under that interpretation, Meno's argument demonstrates the impossibility of a certain kind of knowledge. In what follows we will consider Meno's argument in more detail, and we will try to show that similar arguments are available for many other conceptions of knowledge. T h e modern M e n o arguments reveal a diverse and intricate structure in the theories of knowledge and of inquiry, a structure whose exploration has just begun. While we will attempt to show that our reading of the argument fits reasonably well

3

4

INFERENCE AND M E T H O D

w i t h P l a t o ' s text, w e d o n o t a i m to a r g u e a b o u t Plato's i n t e n t . I t is e n o u g h t h a t the t r a d i t i o n a l text c a n be e l a b o r a t e d i n t o a systematic a n d c h a l l e n g i n g s u b j e c t of c o n t e m p o r a r y i n t e r e s t . 2

2. T H E M E N O In o n e passage in t h e Meno, to a c q u i r e k n o w l e d g e is to a c q u i r e a t r u t h t h a t c a n be given a special logical form. T o a c q u i r e k n o w l e d g e of v i r t u e is to c o m e to k n o w a n a p p r o p r i a t e t r u t h t h a t states a c o n d i t i o n , or c o n j u n c t i o n of c o n d i tions, necessary a n d sufficient for a n y i n s t a n c e of virtue. P l a t o ' s Socrates will not a c c e p t lists, or d i s j u n c t i v e c h a r a c t e r i z a t i o n s . Socrates: I seem to be in great luck, Meno; while I am looking for one virtue, I have found you to have a whole swarm of them. But, Meno, to follow up the image of swarms, if I were asking you what is the nature of bees, and you said that they are many and of all kinds, what would you answer if I asked you: "Do you mean that they are many and varied and different from one another in so far as they are bees? Or are they no different in that regard, but in some other respect, in their beauty, for example, or their size or in some other such way?" Tell me, what would you answer if thus questioned? Meno: I would say that they do not differ from one another in being bees. Socrates: Suppose I went on to say: "Tell me, what is this very thing, Meno, in which they are all the same and do not differ from one another?" Would you be able to tell me? Meno: I would. Socrates: The same is true in the case of the virtues. Even if they are many and various, all of them have one and the same form which makes them virtues, and it is right to look to this when one is asked to make clear what virtue is. Or do you not understand what I mean? T h e r e is s o m e t h i n g p e c u l i a r l y m o d e r n a b o u t the Meno. T h e s a m e rejection of d i s j u n c t i v e c h a r a c t e r i z a t i o n s c a n be f o u n d in several c o n t e m p o r a r y a c c o u n t s of e x p l a n a t i o n . 3 W e m i g h t say t h a t Socrates requires t h a t M e n o p r o d u c e a n a p p r o p r i a t e a n d t r u e u n i v e r s a l b i c o n d i t i o n a l sentence, in w h i c h a p r e d i c a t e signifying 'is v i r t u o u s ' flanks o n e side of the b i c o n d i t i o n a l , a n d a c o n j u n c t i o n of a p p r o p r i a t e p r e d i c a t e s o c c u r s o n the o t h e r side of the b i c o n d i t i o n a l . L e t us so say. N o t h i n g is lost b y t h e a n a c h r o n i s m a n d , as we shall see, m u c h is g a i n e d . S t a t e m e n t s of e v i d e n c e also h a v e a logical f o r m in t h e Meno. W h e t h e r the topic is bees, o r v i r t u e , o r g e o m e t r y , t h e evidence Socrates considers consists of instances a n d n o n - i n s t a n c e s of v i r t u e , of g e o m e t r i c properties, or w h a t e v e r the topic m a y be. E v i d e n c e is s t a t e d in t h e singular. T h e task of a c q u i r i n g k n o w l e d g e thus assumes the following f o r m . O n e is p r e s e n t e d w i t h , o r finds, in w h a t e v e r w a y , a series of e x a m p l e s a n d n o n e x a m p l e s of the f e a t u r e a b o u t w h i c h o n e is i n q u i r i n g , a n d f r o m these e x a m p l e s a true, u n i v e r s a l b i c o n d i t i o n a l w i t h o u t d i s j u n c t i o n s is to be p r o d u c e d . I n the

5

THOROUGHLY MODERN MENO

Meno that is not e n o u g h for k n o w l e d g e to h a v e been a c q u i r e d . T o a c q u i r e k n o w l e d g e it is insufficient to p r o d u c e a truth of the r e q u i r e d f o r m ; one must also know that o n e has p r o d u c e d a truth. W h a t c a n this r e q u i r e m e n t m e a n ? S o c r a t e s a n d M e n o a g r e e in distinguishing k n o w l e d g e f r o m m e r e true opinion, a n d they a g r e e that k n o w l e d g e requires at least true o p i n i o n . M e n o thinks the d i f f e r e n c e b e t w e e n k n o w l e d g e a n d true o p i n i o n lies in the g r e a t e r

reliability

of k n o w l e d g e , b u t Socrates insists that true o p i n i o n c o u l d , b y a c c i d e n t as it w e r e , be as r e l i a b l e as k n o w l e d g e : Meno: . . . But the man who has knowledge will always succeed, whereas he who has true opinion will only succeed at times. Socrates: How do you mean? Will he who has the right opinion not always succeed, as long as his opinion is right? Meno: That appears to be so of necessity, and it makes me wonder, Socrates, this being the case, why knowledge is prized far more highly than right opinion, and why they are different. S o c r a t e s a n s w e r s each question, after a fashion. T h e d i f f e r e n c e b e t w e e n knowledge a n d true opinion is in the special tie, the b i n d i n g c o n n e c t i o n , between w h a t the proposition is a b o u t a n d the fact o f its belief. A n d opinions that a r e tied in this special w a y a r e not only reliable, they a r e l i a b l e to stay, a n d it is that w h i c h m a k e s them especially prized: Socrates: T o acquire an untied work of Daedalus is not worth much, like acquiring a runaway slave, for it does not remain, but it is worth much if tied down, for his works are very beautiful. What am I thinking of when I say this? True opinions. For true opinions, as long as they remain, are a fine thing and all they do is good, but they are not willing to remain long, and they escape from a man's mind, so that they are not worth much until one ties them down by an account of the reason why. And that, Meno my friend, is recollection, as we previously agree. After they are tied down, in the first place they become knowledge, and then they remain in place. That is why knowledge is prized higher than correct opinion, and knowledge differs from correct opinion in being tied down. P l a t o is c h i e f l y c o n c e r n e d with the d i f f e r e n c e b e t w e e n k n o w l e d g e a n d true o p i n i o n , a n d o u r c o n t e m p o r a r i e s h a v e f o l l o w e d this interest. T h e recent focus of e p i s t e m o l o g y has been the special intentional a n d c a u s a l s t r u c t u r e required for k n o w i n g . B u t M e n o ' s a r g u m e n t does not d e p e n d on the details of this analysis; it d e p e n d s , instead, on the c a p a c i t y f o r true o p i n i o n that the c a p a c i t y to a c q u i r e k n o w l e d g e implies. T h a t is the c a p a c i t y to find the truth of a question, to r e c o g n i z e it w h e n f o u n d , to stick w i t h it a f t e r it is f o u n d , a n d to d o so whatever the truth m a y be. S u p p o s e that Socrates could meet M e n o ' s r h e t o r i c a l c h a l l e n g e a n d recognize the truth w h e n he met it: w h a t is it he w o u l d then b e a b l e to do? S o m e thing like the f o l l o w i n g . I n e a c h of m a n y d i f f e r e n t imaginable

(we d o not say

possible s a v e in a logical sense) c i r c u m s t a n c e s , in w h i c h distinct c l a i m s a b o u t

6

INFERENCE AND M E T H O D

virtue (or w h a t e v e r ) a r e true, u p o n receiving e n o u g h evidence, a n d consideri n g e n o u g h hypotheses, Socrates w o u l d hit u p o n the right hypothesis a b o u t virtue for that possible circumstance, a n d w o u l d then (and only then) a n n o u n c e t h a t the correct hypothesis is indeed correct. N e v e r m i n d just how Socrates w o u l d b e a b l e to d o this, b u t agree that, if he is in the a c t u a l c i r c u m stance c a p a b l e o f c o m i n g to k n o w , then that c a p a c i t y implies the c a p a c i t y j u s t stated. K n o w l e d g e requires the ability to c o m e to believe the truth, to recognize w h e n o n e believes the truth (and so to be able to continue to believe the truth), a n d to d o so w h a t e v e r the true state of affairs m a y be. S o u n d e r s t o o d , M e n o ' s a r g u m e n t is valid, or at least its premises c a n be plausibly e x t e n d e d to form a valid a r g u m e n t for the impossibility o f k n o w l edge. T h e l a n g u a g e o f possible worlds is convenient for stating the a r g u m e n t . Fix some list o f predicates V , P I , . . . , Pn, and consider all possible w o r l d s (with c o u n t a b l e d o m a i n s ) t h a t assign extensions to the predicates. In some of these worlds there will be true universal biconditional sentences with V o n one side a n d c o n j u n c t i o n s o f some o f the Pi or their negations on the other side. T a k e pieces o f e v i d e n c e a v a i l a b l e from a n y one o f these structures to be increasing c o n j u n c t i o n s a t o m i c o r negated atomic formulas simultaneously satisfiable in the structure. L e t Socrates receive an u n b o u n d e d sequence of singular sentences in this v o c a b u l a r y , so that the sequence, if continued, will e v e n t u a l l y include e v e r y a t o m i c or negated atomic formula (in the v o c a b u l a r y ) that is satisfiable in the structure. L e t co range over worlds. W i t h M e n o , as w e h a v e read him, say t h a t Socrates c a n c o m e to know a sentence, S, of the a p p r o p r i a t e f o r m , true in w o r l d (0, only if (i) for e v e r y possible sequence o f presentation of evidence from w o r l d a> Socrates e v e n t u a l l y a n n o u n c e s that S is true, and (ii) in e v e r y w o r l d , a n d for every sequence from that w o r l d , if there is a sentence of the a p p r o p r i a t e form true in that w o r l d , then Socrates c a n e v e n t u a l l y consider some true sentence of the a p p r o p r i a t e form in that w o r l d , c a n a n n o u n c e that it is true in that world (while never m a k i n g such a n a n n o u n c e m e n t of a sentence that is not true in that w o r l d ) , and (iii) in e v e r y w o r l d , a n d for every sequence from that w o r l d , if no sentence o f the a p p r o p r i a t e f o r m is true in the w o r l d , then Socrates refrains from a n n o u n c ing of a n y sentence of that form that it is true. M e n o ' s a r g u m e n t is n o w a piece of mathematics, and it is s t r a i g h t f o r w a r d to p r o v e that he is correct: no matter w h a t powers w e i m a g i n e Socrates to h a v e , he c a n n o t a c q u i r e k n o w l e d g e , provided " k n o w l e d g e " is understood to entail these requirements. N o hypotheses a b o u t the causal conditions for k n o w l e d g e defeat the a r g u m e n t unless they defeat the premises. Skepticism need not rest on e m p i r i c a l reflections a b o u t the weaknesses o f the h u m a n mind. T h e impossibility o f k n o w l e d g e c a n be demonstrated a priori. W h a t e v e r s e q u e n c e of evi-

THOROUGHLY MODERN

MENO

7

dence Socrates m a y receive that agrees with a hypothesis of the required form, there is some structure in which that evidence is true but the hypothesis is false; so that if at any point Socrates announces his conclusion, there is s o m e i m a g i n a b l e circumstance in which he will be wrong. We should note, however, that in those circumstances in which there is no truth of the required form, Socrates c a n eventually c o m e to know that there is no such truth, provided he has an initial, finite list of all of the predicates that m a y occur in a definition. H e can a n n o u n c e with perfect reliability the a b s e n c e of any purely universal conjunctive characterizations of virtue if he has received a c o u n t e r e x a m p l e to every h y p o t h e s i s — a n d if the n u m b e r of predicates are finite, the n u m b e r of hypotheses will be finite, and if no hypothesis of the required form is true, the counterexamples will eventually occur. If the relevant list of predicates or properties were not provided to S o c r a t e s initially, then he could not know that there is no knowledge of a subject to be had. 3. W E A K E N I N G

KNOWLEDGE

Skepticism has an ellipsis. T h e content of the d o u b t that knowledge is possible depends on the requisites for knowledge, and that is a m a t t e r over which philosophers dispute. R a t h e r than supposing there is one true account of knowledge to be given, if only philosophers could find it, our disposition is to inquire a b o u t the possibilities. O u r notion of knowing is surely v a g u e in ways, and there is room for more than one interesting doxastic state. About the conception of knowledge we have extracted from M e n o there is no d o u b t as to the Tightness of skepticism. N o one can have that sort of knowledge. Perhaps there a r e other sorts that can be h a d . We could restrict the set of possibilities that must be considered, eliminating most of the possible worlds, and m a k e requirements (i), (ii), and (iii) apply only to the reduced set o f possibilities. We would then have a revised conception of knowledge that requires only a reduced scope, as we shall call the r a n g e of structures over which Socrates, or you or we, must succeed in order to be counted as a knower. T h i s is a recourse to which we will have eventually to come, but let us put it aside for now, a n d consider instead what might otherwise be d o n e a b o u t weakening conditions (i), (ii), a n d (iii). Plato's Socrates emphasizes this difference between knowledge and m e r e true opinion: knowledge stays with the knower, but mere o p i n i o n , even true opinion, m a y flee a n d be replaced by falsehood or want of opinion. T h e evident thing to consider is the requirement that for Socrates to c o m e to know the truth in a certain world, Socrates be able to find the truth in each possible world, a n d never a b a n d o n it, but not be obliged to a n n o u n c e that the truth has been found when it is found. Whatever the relations of cause a n d intention that knowledge requires, surely M e n o requires too m u c h . H e requires, as we have reconstructed his a r g u m e n t , that we come to believe through a reliable proce-

INFERENCE AND METHOD

8

d u r e , a p r o c e d u r e or c a p a c i t y t h a t w o u l d , w e r e the w o r l d d i f f e r e n t , l e a d to a p p r o p r i a t e l y d i f f e r e n t c o n c l u s i o n s in t h a t c i r c u m s t a n c e . B u t M e n o also req u i r e s t h a t w e k n o w w h e n the p r o c e d u r e has s u c c e e d e d , a n d t h a t seems m u c h like d e m a n d i n g t h a t w e k n o w t h a t w e k n o w w h e n w e k n o w . K n o w i n g t h a t w e k n o w is a n a t t r a c t i v e p r o p o s i t i o n , b u t it d o e s not seem a prerequisite for k n o w l e d g e , or if it is, then b y the p r e v i o u s a r g u m e n t , k n o w l e d g e is impossible. I n e i t h e r case, the properties o f a w e a k e r c o n c e p t i o n o f k n o w l e d g e d e s e r v e o u r study. T h e idea is t h a t S o c r a t e s c o m e s e v e n t u a l l y to e m b r a c e the truth a n d to stick w i t h it in e v e r y case, a l t h o u g h h e does not k n o w at w h a t point h e has succ e e d e d : he is n e v e r sure t h a t h e will not, in the f u t u r e , h a v e to c h a n g e his hypothesis. In this c o n c e p t i o n o f k n o w l e d g e , there is n o m a r k o f success. W e m u s t then think o f S o c r a t e s as c o n j e c t u r i n g the truth forever. S i n c e S o c r a t e s d i d not live forever, n o r shall w e , it is better to think o f S o c r a t e s as h a v i n g a procedure t h a t c o u l d b e a p p l i e d indefinitely, e v e n w i t h o u t the l i v i n g S o c r a t e s . T h e p r o c e d u r e has m a t h e m a t i c a l properties t h a t S o c r a t e s does not. F o r S o c r a t e s to k n o w t h a t S in w o r l d co in w h i c h S is true n o w implies t h a t S o c r a t e s ' b e h a v i o r a c c o r d s w i t h a p r o c e d u r e w i t h the f o l l o w i n g properties: (i*) for e v e r y possible s e q u e n c e o f e v i d e n c e f r o m w o r l d QJ, after a finite s e g m e n t is p r e s e n t e d , the p r o c e d u r e c o n j e c t u r e s S e v e r after, a n d (ii*) for e v e r y possible s e q u e n c e o f e v i d e n c e f r o m a n y possible w o r l d , if a s e n t e n c e o f the a p p r o p r i a t e f o r m is true in that w o r l d , then after a finite segm e n t o f the e v i d e n c e is presented the p r o c e d u r e c o n j e c t u r e s a true s e n t e n c e of t h e a p p r o p r i a t e f o r m e v e r after. T h e s e c o n d i t i o n s c e r t a i n l y are not sufficient for a n y d o x a s t i c state v e r y close to o u r o r d i n a r y n o t i o n o f k n o w l e d g e , since S o c r a t e s ' b e h a v i o r m a y in the a c t u a l w o r l d a c c o r d w i t h a p r o c e d u r e satisfying (i*) a n d (ii*) e v e n w h i l e S o c r a t e s lacks the disposition to a c t in a c c o r d with the p r o c e d u r e in other-circumstances. F o r k n o w l e d g e , S o c r a t e s must h a v e such a disposition. B u t he c a n o n l y h a v e s u c h a disposition if there exists a p r o c e d u r e m e e t i n g c o n d i t i o n s (i*) a n d (ii*). Is there? I f the logical f o r m o f w h a t is to be k n o w n is restricted to universal b i c o n d i t i o n a l s o f the sort P l a t o r e q u i r e d , then there is i n d e e d such a p r o c e d u r e . I f S o c r a t e s is u n a b l e to a c q u i r e this sort o f k n o w l e d g e , then it is b e c a u s e of p s y c h o l o g y or s o c i o l o g y or b i o l o g y , not in v i r t u e o f m a t h e m a t i c a l impossibilities. S k e p t i c i s m a b o u t this sort o f k n o w l e d g e c a n n o t be a priori. T h e r e is no g e n e r a l a r g u m e n t o f M e n o ' s kind against the possibility o f a c q u i r i n g this sort o f knowledge. T h e w e a k e n i n g o f k n o w l e d g e m a y be u n - P l a t o n i c , b u t it is not u n p h i l o s o p h i c a l . F r a n c i s B a c o n ' s Novum Organum describes a p r o c e d u r e t h a t w o r k s for this case, a n d his c o n c e p t i o n o f k n o w l e d g e seems r o u g h l y to a c c o r d w i t h it. J o h n S t u a r t M i l l ' s c a n o n s o f m e t h o d are, o f course, s i m p l y p i r a t e d f r o m B a -

T H O R O U G H L Y MODERN MENO

9

c o n ' s m e t h o d . H a n s R e i c h e n b a c h used n e a r l y the s a m e c o n c e p t i o n o f k n o w l e d g e in his " p r a g m a t i c v i n d i c a t i o n " o f i n d u c t i o n , a l t h o u g h he assumed a v e r y d i f f e r e n t logical f o r m for hypotheses, n a m e l y that they are c o n j e c t u r e s a b o u t limits o f relative f r e q u e n c i e s o f properties in infinite sequences. S o w e h a v e a c o n c e p t i o n o f k n o w l e d g e t h a t , at least for s o m e kinds o f h y p o theses, is not s u b j e c t to M e n o ' s p a r a d o x . B u t for w h i c h kinds o f h y p o t h e s e s is this so? W e are not n o w c a p t i v a t e d , if e v e r w e w e r e , b y the n o t i o n t h a t all k n o w l e d g e is d e f i n i t i o n a l in f o r m . P e r h a p s e v e n P l a t o himself w a s not, for the slave b o y learns the t h e o r e m o f P y t h a g o r a s , w h i c h has a m o r e c o m p l i c a t e d logical form. W e a r e interested in o t h e r forms o f hypotheses: positive tests for diseases, a n d tests for their absence; collections o f tests one o f w h i c h will r e v e a l a c o n d i t i o n if it is present. N o r are o u r interests c o n f i n e d to single h y p o t h e s e s considered i n d i v i d u a l l y . I f the p r o p e r t y o f b e i n g a s q u a m o u s c a n c e r cell has some c o n n e c t i o n s w i t h o t h e r properties a m e n a b l e to o b s e r v a t i o n , w e w a n t to k n o w all about those c o n n e c t i o n s . W e w a n t to d i s c o v e r the w h o l e theory a b o u t the s u b j e c t m a t t e r , or as m u c h as w e c a n of it. W h a t w e m a y wish to d e t e r m i n e , then, is w h a t classes o f theories c a n c o m e to be k n o w n a c c o r d i n g to o u r w e a k e r c o n c e p t i o n o f k n o w l e d g e . H e r e , as w e use the n o t i o n o f t h e o r y , it m e a n s the set of all true c l a i m s in s o m e f r a g m e n t o f l a n g u a g e . W a n t i n g to k n o w the truth a b o u t a p a r t i c u l a r question is then a special case, since the q u e s t i o n c a n be f o r m u l a t e d as a c l a i m a n d its d e n i a l , a n d the p a i r f o r m a f r a g m e n t of l a n g u a g e w h o s e true claims a r e to be d e c i d e d . W h a t w e wish to d e t e r m i n e is w h e t h e r all o f w h a t is true a n d c a n be stated in s o m e f r a g m e n t o f l a n g u a g e c a n be k n o w n . E i t h e r the possibility o f k n o w l e d g e d e p e n d s on the f r a g m e n t of l a n g u a g e considered or it d o e s not. If it does, then m a n y distinct f r a g m e n t s o f l a n g u a g e m i g h t be of the sort that p e r m i t k n o w l e d g e o f w h a t c a n be said in t h e m , a n d the classification o f f r a g m e n t s that d o , a n d that d o not, p e r m i t such k n o w l e d g e b e c o m e s a n interesting task. F o r w h i c h f r a g m e n t s o f l a n g u a g e , if a n y , a r e there valid a r g u m e n t s o f M e n o ' s sort against the possibility o f k n o w l e d g e , a n d for w h i c h f r a g m e n t s a r e there not? T h e s e are s t r a i g h t f o r w a r d m a t h e m a t i c a l questions, a n d their answers, or some o f their answers, are as follows: C o n s i d e r a n y first-order l a n g u a g e ( w i t h o u t identity) in w h i c h all p r e d i c a t e s are m o n a d i c , and there a r e no s y m b o l s taken to represent functions. T h e n a n y true t h e o r y in s u c h a l a n g u a g e c a n be l e a r n e d , or at least there are n o v a l i d M e n o a n a r g u m e n t s a g a i n s t such k n o w l e d g e . If the l a n g u a g e is m o n a d i c b u t w i t h i d e n t i t y , or if the l a n g u a g e c o n t a i n s a p r e d i c a t e that is not m o n a d i c , then neither the f r a g m e n t that consists o n l y of u n i v e r s a l l y q u a n t i f i e d f o r m u l a s , nor the f r a g m e n t that consists o n l y o f existentially q u a n t i f i e d f o r m u l a s , nor a n y p a r t o f the l a n g u a g e c o n t a i n i n g e i t h e r of these f r a g m e n t s , is such that e v e r y true theory in these f r a g m e n t s c a n

be

known. In e a c h of the l a t t e r cases a n a r g u m e n t o f M e n o ' s kind c a n be c o n s t r u c t e d to show that k n o w l e d g e is impossible.

10

I N F E R E N C E AND M E T H O D

4. T I M E S FOR A L L T H I N G S The weakened conception of knowledge is still very strong in at least one respect. It requires for the possibility of knowledge of an infinite wealth of claims that there be a time at which all of them are known—that is, a single time after which all and only the truths in a fragment of language are conjectured. We might instead usefully consider the following circumstance: When investigating hypotheses in a fragment of language, Socrates is able, for each truth, eventually to conjecture it and never subsequently to give it up; and Socrates is also able, for each falsehood, eventually not to conjecture it and never after to put it forward. Plato's Socrates illustrates that the slave boy can "recollect" the Pythagorean theorem from examples and appropriate questions, and presumably in Plato's view the slave boy could be made to recollect any other truth of geometry by a similar process. But neither the illustration nor the view requires that the slave boy, or anyone else, eventually be able to recollect the whole of geometry. There may be no time at which Socrates knows all of what is true and can be stated in a given fragment of language. Yet the disposition to follow a procedure that will eventually find every truth and eventually avoid every falsehood is surely of fundamental interest to the theory of knowledge. Call a procedure that has the capacity to converge to the whole truth at some moment, as in the discussion of the previous section, an EA learning procedure, and call an A E learner a procedure that for each truth has the capacity to converge to that truth by some moment, and for each falsehood avoids it ever after some moment. Every EA learner is an A E learner, but is the converse true? Or more to the point, are there fragments of language for which there are AE procedures but no EA procedures? There are indeed. Consider the set of all universal sentences, with identity, and with any number of predicates of any arity and any number of function symbols of any arity. By the negative result stated previously, there is no EA procedure for that fragment of language, no procedure that, for every (countable) structure, and every way of presenting the singular facts in the structure, will eventually conjecture the theory (in the language fragment) true in that structure. But there is an AE procedure for this fragment. If, for knowledge about a matter, Socrates is required only to have a disposition to follow an AE procedure for the language of the topic, then no Menoan argument shows that Socrates cannot acquire knowledge, even if Socrates does not know the relevant predicates or properties beforehand. The improvement does not last. If we consider the fragment of language that allows up to one alternation of quantifiers, whether from universal to existential or from existential to universal, it again becomes impossible to acquire knowledge; there are no A E procedures for this fragment that are immune from arguments of Meno's kind.

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5. DISCOVERY AND SCOPE Whether we consider EA discovery or AE discovery, we soon find that arguments of Meno's kind succeed. T h e same sort of results obtain if we further weaken the requirements for knowledge. We might, for example, abandon Plato's suggestion that when a truth is known it is not subsequently forgotten or rejected. We might then consider the requirement that Socrates be disposed to behave in accordance with a procedure that, as it considers more and more evidence about a question, is wrong in its conjectures only finitely often, is correct infinitely often, but may also suspend judgment infinitely often. Osherson and Weinstein have shown that even with this remarkably weak conception there are questions that cannot, in senses parallel to those above, be known. O r we might allow various sorts of approximate truth; for many of them, arguments parallel to Meno's are available. T h e conceptions of knowledge we have discussed place great emphasis on reliability. T h e y demand that we not come to our true beliefs by chance but in accordance with procedures that would find the truth no matter what it might be, so long as the procedures could be carried out. What the Meno arguments show is that in the various senses considered, for most of the issues that might invite discovery, procedures so reliable do not exist. The antiskeptical response ought to be principled retreat. In the face of valid arguments against the possibility of procedures so reliable, and hence against the possibility of corresponding sorts of knowledge, let us consider procedures that are not so reliable, and regard the doxastic state that is obtained by acting in accord with them as at least something better and more interesting than accidental true belief. For each of the requirements on knowledge considered previously, and for others, we can ask the following kind of question: For each fragment of language, what are the classes of possible worlds for each of which there exists a procedure that will discover the truths of that fragment for any world in the class? T h e question may be too hard to parse. Let us define it in pieces. Let a discovery problem be any (recursive) fragment F of a formal language, together with a class K of countable relational structures for that fragment. O n e such class K is the class of all countable structures for the language fragment, but any subsets of this class may also be considered. A discovery procedure for the discovery problem is any procedure that, for every k in K and every presentation of evidence from k, "converges" to all of the sentences in F that are true in k. "Convergence" may be in the EA sense, the AE sense, or some other sense altogether (such as the weak convergence criterion considered two paragraphs previously). What the results we have described tell us is that for many fragments F, if K is the set of all countable structures for F, then there are no discovery procedures for pairs . T h a t does not imply that there are no discovery proce-

12

INFERENCE AND M E T H O D

dures for pairs rt U -O - -S — c O S « J bC o o o . C * S — o — >s V V V2 s E 5 ai If a i
). T h a t is, we are interested in P s o that B|G| — P

and

V £ {PD, P R ^ , . . . , P R n }

where P D a n d P R j universally express Q D and Q R j respectively. I call P a n empirical procedure expression and the set B a n empirical base for P. This corresponds to the customary distinction between "intensional" and "extensional" d a t a base relations ([23], 100). V. THEORY ELEMENTS V . l . Introduction. In this section I will sketch how procedure languages may be used to provide syntactical expression for the essential model-theoretic entities appearing in structuralist reconstructions of empirical theories ([1]. [17]). These include potential models and models (sec. V.2), partial potential models (sec. V.3), and constraints (sec. V.4). I will indicate how questions of eliminability and definability that elude precise semantic formulation might be handled with this syntax. I do not consider the structuralist concept of intertheoretical link here, but I believe one might extend procedural syntax to this as well. V.2. Empirical Laws. Clearly, M (P, Q J could be used to characterize empirically interesting classes of models. However, Q is a semantic entity. M o r e appropriate, for our purposes, is a purely syntactic characterization. Consider pairs of empirical procedure expressions .

M e m b e r s of M p p [ U , n ] a r e k 4- 1, . . . , kl " r e d u c t s " of m e m b e r s of M p [ U , n, t]. W e call m e m b e r s of M p p [ U , n ] partial potential models. I n t u i t i v e l y , t h e relations in the places k + 1 , . . . , 1 a r e theoretical relations a n d those in places 1, . . . , k a r e nontheoretical relations. M p [ U , n, t] is the class of theoretical structures w h i l e M p p [ U , n ] is t h e class of nontheoretical structures. V.3.1.2. D e n o t e the " R a m s e y f u n c t o r " ( [ 1 ] , sec. I I . 4 ) f r o m M p o n t o M p p by R a m : M p [ U , n, t]

M p p [ U , n]

so that, for all m p = in M p , R a m ( m p ) = . Q u e r i e s for M p [ U , n, t] a n d M p p [ U , n] we will d e n o t e respectively b y Q.p:Mp[U,n,t]^R[U]

and

Q , p p : M p p [ U , n] ^

R[U],

Y.3.1.3. Expression p a i r s < L p , L p ' > for M p d e t e r m i n e sets of M p - m o d e l s in t h e m a n n e r described a b o v e . V i a the R a m s e y f u n c t o r , they also d e t e r m i n e sets

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of M p p -models: Ram(M(Lp,Lp')). Similarly, a theory L for M p determines a set of M p p -models which we have called the nontheoretical content of the theory element and testing whether L p ( m p p ) £ L p ' ( m p p ) . T h e first successful test is sufficient to legitimate m p p as a member of the nontheoretical content of the theory whose single theoretical law is ( L p , L p '>.

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V . 3 . 3 . Eliminability and Definability. P r o c e d u r a l syntax permits us to raise questions of eliminability a n d definability of theoretical concepts in a precise w a y . M o r e i m p o r t a n t , it m a y prove possible, using this a p p a r a t u s , to address these questions for theories that a r e not readily f o r m a l i z a b l e in first-order logic. V.3.3.2. Given interpreted p r o c e d u r e l a n g u a g e < | L ( G ) , I> a n d t h e theory element < M p p [ U , n], M p [ U , n, t], M ( L p ) > with the m e m b e r s of L p in |P, we say t h a t the theory L p p = , . . . , < L p p i , L p p i ) > for M p p with all L p p . e | P is Ramsey equivalent to L iff M(Lpp) =

R^(M(L)).

W e then ask, Is the following true? R A M S E Y E L I M I N A B I L I T Y T H E S I S : For all theory elements < M p p , M p , M ( L p ) > , there is a theory L p p for M p p so t h a t L p p is R a m s e y equivalent to L. V.3.3.3. T h e question of definability of theoretical terms a n d its relation to model-theoretic eliminability m a y be raised in this w a y . S u p p o s e {Pi

Pn}|G| - P

a n d relative to the interpreted p r o c e d u r e l a n g u a g e < | L ( G ) , I ) , M(PJQ)eflM(P„Qi) i then we say, relative to < | L ( G ) , I ) , Q, definable in terms of { Q , i , . . . , (¿„} in M(P, QJ. Note t h a t 'definability' applies to queries r a t h e r t h a n to expressions for them. In the special case of queries Q R j we say t h a t Rk is definable in terms of {/?;,,..., in M £ M p [ U , r] w h e n t h e r e exist Pk a n d {Pj.} {Pis} | G | - Pk and M =

M ( P k > Q R k ) s n jM ( P i J . Q . R « l ) -

T h u s , for theory element < M p p [ U , n ] , M p [ U , n, t], M ( L p ) ) we m a y ask w h e t h e r R k , k 6 t, is definable in terms of {R^}, i, j e n, in M ( L p ) . Clearly, the answer to both eliminability a n d definability questions d e p e n d s on | L. Y.4. Constraints Y.4.1. Constraints and n-ary Queries. Model-theoretically, a c o n s t r a i n t is just a subset of P o ( M p ) — t h e set of all subsets of M p t h a t satisfy the " c o n s t r a i n t . " A simple e x a m p l e of a constraint is the r e q u i r e m e n t t h a t identical particles in different models of classical particle m e c h a n i c s h a v e t h e s a m e mass values. This e x a m p l e is trivial in t h a t it is equivalent to a r e q u i r e m e n t on the u n i o n of t h e constrained m p 's. However, examples of nontrivial constraints such as the "extensivity c o n s t r a i n t " on mass in particle m e c h a n i c s a b o u n d in empirical

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science. T h a t representing nontrivial constraints is essential to representing empirical theories has been argued in detail in [1 ], section II.2. V.4.1.2. C a n procedural syntax be used to represent constraints? First, let us restrict our attention to constraints that are sets of two-member subsets of M p [ U , r]. W e might think of a binary queries as functions: Q.:Mp[U,r] x M p [ U , r ] ^ R [ U ] so t h a t , f o r all < m p , m p ' > e M p x M p , Q,(m p , m p ') e R [ U , (D(m p ) u D ( m p ' ) ) ] . T h a t is, Q_maps pairs of potential models into relations on the union of their domains. A computable query of this type is (partial) recursive and consistent in a natural extension of the sense of section I I I above. V.4.1.3. We may consider a binary interpretation for procedures in language |L as a function: I P m : | P x M p [ U , r ] x M p [ U , r] -» S E T ( R [ U ] m , R [ U ] ) u { J.}. Relative to a binary interpretation, P expresses the binary query Qin (mp, m p ') iff IP n + 1 (P, m p , m p ') (m p , m p ') = Q,(m p , m p ' ) . For P in | P and binary query Q,we may define K (P, Q) = { < m P . m P ' > I p expresses Q i n may be used in problem solving essentially as extensions to the formation rules G. They provide additional "substantive" principles for constructing new procedures from given procedures. VI.2.2. T h e basic idea of using that o p e r a t e on the full theoretical structures. W e view these procedures as constructed, in part, from " g e n e r a t o r s " r a t h e r t h a n q u e r y procedures for theoretical concepts (sec. V . 3 . 2 ) . Generally, p r o b l e m solving here is m o r e c o m p l e x just because there m a y be multiple values g e n e r a t e d for theoretical concepts that m a k e L p m p ) £ L p ' ( m p ) . T h e process o f p r o b l e m solving using the formation rules and laws in the m a n n e r sketched a b o v e will be essentially the same. However, w e should expect the S arrived at to yield arrays o f solutions corresponding to the multiple possible values o f the theoretical concepts. V I . 3 . 2 . Consider next the case in which the " d a t a " queries are j u s t those for the nontheoretical c o n c e p t s Q _ = < Q R ! , . . . , Q R n > and the " u n k n o w n " query is theoretical Q R t . T h i s is a p r o b l e m in d e t e r m i n i n g the value o f the theoretical c o n c e p t R , from c o m p l e t e d a t a a b o u t n o n t h e o r e t i c a l concepts. T h e S that solves this p r o b l e m will g e n e r a l l y not yield unique solutions for specific values o f Q,. B u t there m a y be s o m e m p ' s in which the solution is unique. T h e s e correspond to systems t h a t provide " m e a s u r e m e n t " methods for R , . V I . 3 . 2 . U p to this point we h a v e viewed problems and p r o b l e m solving as having to do with a single m e m b e r o f M p [ U , r]. I t has been argued at length in

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[17], chapters 4 and 5, and in [1], section II.2, that this is an incomplete a n d seriously i n a d e q u a t e picture of the way empirical science is practiced. Problem solving essentially involves the " i m p o r t a n d e x p o r t " of information across different models of the same theory. A procedural version of this view is roughly this. T h e generators for values of theoretical concepts are replaced (supplemented) by procedures for " i m p o r t i n g " these values f r o m other m e m bers of M p [ U , r ] — t h o s e in which the laws of the theory suffice to d e t e r m i n e them uniquely. This suggests that, in practice, in "real-life" situations, the role of generators m a y be relatively insignificant. T h e procedures that effect the " i m p o r t i n g " are essentially constraints on n-tuples of m p 's (sec. V.4). T h i s suggests that a fully a d e q u a t e account of problem solving will require n-ary queries and constraints. VII. T H E O R Y DISCOVERY V I I . 1. Introduction. T h e conception of theory discovery as search is well known ([10]). Procedural syntax provides a precise, general method for bringing this conception to bear on empirical theories represented as model-theoretic structures. In addition it provides a characterization of conceptual i n n o v a t i o n — the discovery of theories employing theoretical concepts—and suggests a way that " s e a r c h " might be expected to yield conceptual innovation. T h i s reformulates and extends the work of Langley et al. in [10]. T h e formulation sketched here opens the way to precisely addressing the question of w h e t h e r there are computational limits on automatic (algorithmic) conceptual innovation. In w h a t follows, I restrict the discussion to discovery of single laws in theories. V I I . 2 . Mpp[U, n]-Data Presentations. T h e conception of law discovery u n d e r consideration is " d a t a - d r i v e n " in the sense that the discovery process is viewed roughly as a function from "data presentations" to "laws." T h e simplest (though not the only interesting) conception of " d a t a " for M p p [ U , n]-structures (relative to an interpreted procedure language < L ( G ) , I > ) is a sequence of d a t a expressions interpreted as M p p [ U , n]-structures: S: N — | D so that ID(S(i))eMpp[U,n], O u r purposes require that we are able to speak partitions of a presentation S into n nonoverlapping parts. I d o this with the formal device of a data partition n: 7i: N

N[n]

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so that, for i, j 8 N [n]; i # j a n d the inverse of n, < n, . V I I . 3 . Nontheoretical Law Discovery. Intuitively, the objective in law discovery is to find the " s t r o n g e s t " law t h a t captures all the d a t a k n o w n a t a n y point in time a n d continues to d o this as m o r e d a t a a r e o b t a i n e d . I n o u r formalism, for d a t a presentation S, we seek a p r o c e d u r e p a i r ( L p p , L p p '> so that, relative to partition n, both: A) for all m, n e

< L p p , L p p '>

n ) - c a p t u r e s S;

B) f o r a l l < P p p , P p p ' ) s o t h a t A ) , M ( L p p , L p p ' >

cM(Ppp,Ppp').

A) requires t h a t all parts of all initial segments of S be c a p t u r e d by < L p p , L p p '>, while B) says < L p p , L p p '> is the strongest p r o c e d u r e p a i r t h a t does A), in the sense t h a t it determines the smallest model class. V I I . 3 . 2 . Search for p r o c e d u r e pairs satisfying A) a n d B) m i g h t simply be conceived as search t h r o u g h a g r a p h || P 2 which is the cross p r o d u c t of the g r a p h of || P = -captures S;

2) at some level, L p p k 's (L p p k "s) have isomorphic parse trees; 3) L p p k 's (L p p k "s) differ only in P k * at isomorphic positions in their parse trees. Property 1) is simply that different nontheoretical laws capture different parts of the initial segment m of the data. However, these different laws have the same form above a certain level; that is, they use data processed at a lower level in the same way—2). They differ only in the way presentation d a t a is processed P k * at lower levels—3). VII.4.3. Intuitively, the different P k *'s correspond to different ways of measuring the value of the value of the same theoretical concept in different

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m o d e l s of t h e t h e o r y . W e r e c o g n i z e this as the same c o n c e p t j u s t in t h a t t h e o u t p u t of all P k *'s is used in t h e s a m e w a y b y t h e laws. I n fact, t h e r e is really j u s t one l a w — o n c e w e h a v e " i d e n t i f i e d " t h e d i f f e r e n t P k *'s as o u t p u t t i n g values of the same c o n c e p t . V e r y r o u g h l y , w h a t I a m suggesting h e r e is a w a y of t u r n i n g o l d - f a s h i o n e d " o p e r a t i o n a l i s m " a r o u n d . I n s t e a d of i d e n t i f y i n g c o n c e p t s by t h e i r m e t h o d s of d e t e r m i n a t i o n , I suggest w e identify t h e m by the f u r t h e r use we m a k e of t h e results of ( d i f f e r e n t ) d e t e r m i n a t i o n m e t h o d s . N o t e t h a t it is the p r o c e d u r a l s y n t a x t h a t m a k e s it possible to identify precisely the " u s e s " of results. V I 1.4.4. I sketch h e r e a process t h a t m i g h t be t u r n e d i n t o a n a l g o r i t h m for d i s c o v e r i n g t h e simplest k i n d of theoretical l a w — t h a t c o n t a i n i n g only o n e theoretical c o n c e p t . A s s u m i n g o n e thinks t h a t such discoveries c o u n t as " i n t e r e s t i n g " c o n c e p t u a l i n n o v a t i o n in e m p i r i c a l science, t w o kinds of things m i g h t be d o n e with such a sketch. First, o n e m i g h t try to i m p l e m e n t the sketch in s o m e w o r k i n g p r o c e d u r a l l a n g u a g e a n d see h o w it f a r e d o n s o m e n o n t r i v i a l e x a m ples (e.g., m o m e n t u m m e c h a n i c s ) . S e c o n d , o n e m i g h t try to c h a r a c t e r i z e the process in s o m e m o r e a b s t r a c t w a y a n d investigate its c o m p u t a t i o n a l p r o p erties. Success with t h e first w o u l d s h o w t h a t interesting c o n c e p t u a l i n n o v a t i o n c a n be a u t o m a t e d . F a i l i n g this, o n e m i g h t p u r s u e t h e second line in the h o p e of s h o w i n g t h a t the kind of a l g o r i t h m n e e d e d for c o n c e p t u a l i n n o v a t i o n is c o m p u t a t i o n a l l y " h a r d " ( [ 9 ] , c h a p . 13), thus p r o v i d i n g a kind of " i m p o s s i b i l i t y " result. It is t h e need to r e p r e s e n t a n d search (intelligently) t h r o u g h d a t a p a r t i tions t h a t suggests this m i g h t be the case.

VIII.

APPENDIX

A. 1. In the case of b i n a r y r e l a t i o n a l s t r u c t u r e s M p [ U , ], the " c o n v e r s e " function: Q:Mp[U,]-R[U] so that, for all m p in M p [ U , ], ^ « D i m p J . R ^ m p ) » = ft, (nip) is in Q J U , ) = R , ( m p ) will also c o m p u t e Q_ in s o m e m e m b e r s of M p [ U , < [ 2 ) ] — n a m e l y , those in which R j is s y m m e t r i c — b u t n o t in o t h e r m e m b e r s . A.2. W e m a y r e p r e s e n t e a c h m p e M p [ U , ] as P R O L O G d a t a base where " f a c t s " of t h e f o r m : dom(a). rel(a, b ) .

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respectively describe D(m p ) and R j (m p ). Taking P R O L O G as our procedure language |L(G), the queries " d o m ( X ) " and "rel(X, Y)" form the empirical basis for theories about binary relational structures. Consider the P R O L O G rules: 1_1 (X):- rel(X, X). 1 _2 (X, Y):- dorn (X), dorn (Y). 1_3 (X, Y):- rel(X,_), rel(_, Y). Via the P R O L O G analog of set-theoretic abstraction (the "findall" function) each of these may be viewed as procedure that computes a function defined on M p [U,] whose value is a set. Procedure pairs formed from these and the basic queries correspond to properties of binary relation structures in the following way: R ,

s D x D

< 1 _ 1 (X), dorn (X) > reflexivity symmetry < 1 _3 (X, Y), rel (X, Y) > transitivity Note that since P R O L O G expressions have both a denotational and a procedural interpretation, the class of models determined by the pair